Class nucmass_ldrop (o2scl)¶

class nucmass_ldrop : public o2scl::nucmass_densmat ¶

Simple liquid drop mass formula.

This class includes bulk, surface and Coulomb contributions to the energy. There is no pairing, no neutron skin, and no isospin contribution to the surface energy.

The NRAPR equation of state from [Steiner05b] is used for the bulk energy by default.

Definition of \( \chi \) and \( n_L \)

The variable \( \chi \) is defined as the fractional volume occupied by and \( n_L \) is the density of nucleons inside the nucleus. If \( V \) is the total volume of the nucleus plus the surrounding Wigner-Seitz cell, then we have

\[ A = V n_L \chi \]

Central densities

Given a saturation density, \( n_0 \) and a transition density, \( n_t \), we set \( I = 1 - 2 Z/A \), and then assume \( \delta = I \). We further assume that the isospin-asymmetric saturation density \( n_L \) is

\[ n_L = n_0 + n_1 \delta^2 \]

and then we can compute \( n_{p} = n_L (1 - \delta)/2 \) and \( n_{n} = n_L (1 + \delta)/2 \) .

Note that \( \delta = I \) implies no neutron skin. A neutron skin occurs when \( \delta < I \), and \( \delta = 0 \) implies a “maximum skin size” which is occurs when no extra neutrons are in center and all extra neutrons are located in the skin, i.e. \( N_{\mathrm{skin}} = N-Z \). See also nucmass_ldrop_skin .

Nuclear radius

The nuclear radius is determined by

\[\begin{eqnarray*} R &=& \left( \frac{3 A}{4 \pi n_L} \right)^{1/3} \end{eqnarray*}\]

Bulk energy contribution

The bulk binding energy contribution ( \( \sim -16 \) MeV per nucleon) and the symmetry energy are computing using the hadronic EOS (either def_had_eos or the EOS specified in the most recent call to set_eos_had_temp_base() ). The bulk energy per baryon is

\[ E_{\mathrm{bulk}}/A = \frac{1}{n_{L} } \left[\varepsilon(n_n,n_p) - n_n m_n - n_p m_p \right] \]

Surface energy contribution

The surface energy density is

\[ \varepsilon_{\mathrm{surf}} = \frac{3 \sigma \chi}{R} \]

where \( \sigma \) is the surface tension. To compute the surface energy per baryon, using \( A = V n_L \chi \), we find

\[ E_{\mathrm{surf}}/A = \frac{3 \sigma}{n_n + n_p} \left[ \frac{3 A}{ 4 (n_n+n_p) \pi} \right]^{-1/3} \]

\[ E_{\mathrm{surf}}/A = \frac{\sigma}{n_L} \left(\frac{36 \pi n_L}{A} \right)^{1/3} \]

where the surface tension \( \sigma \) (in \( \mathrm{MeV}/\mathrm{fm}^2 \)) is given in surften.

Taking a typical value, \( \sigma =1.1~\mathrm{MeV}/\mathrm{fm}^2 \) and \( n_L = 0.16~\mathrm{fm}^{-3} \), gives the standard result, \( E_{\mathrm{surf}}/A = 18~\mathrm{MeV}~A^{-1/3} \) (see nucmass_semi_empirical::Ss in nucmass_semi_empirical).